[Paper Review] On functions taking only prime values

Zhi‐Wei Sun|arXiv (Cornell University)|Feb 29, 2012

Analytic Number Theory Research20 references3 citations

TL;DR

This paper introduces novel number-theoretic functions that generate only prime numbers through modular distinctness conditions on polynomial sequences. It proves that the smallest modulus ensuring pairwise distinct values of expressions like $2k(k-1) \mod m$ yields the least prime greater than $2n-2$, offering a new theoretical characterization of primes via modular arithmetic constraints on quadratic and linear forms.

ABSTRACT

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the set of all prime numbers. For every n=4,5,... we prove that the least prime p>3n with 3|p-1 is just the least positive integer m such that 18k(3k-1) (k=1,...,n) are pairwise distinct modulo m. For d=4,6,12 and n=3,4,...., we prove that the least prime p>2n-2 with p=-1 (mod d) is the smallest integer m such that those (2k-1)^d for k=1,...,n are pairwise distinct modulo m. We also pose several challenging conjectures on primes. For example, we find a surprising recurrence for primes, namely, for every n=10,11,... the (n+1)-th prime p_{n+1} is just the least positive integer m such that 2s_k^2 (k=1,...,n) are pairwise distinct modulo m where s_k = sum_{j=1}^k(-1)^{k-j}p_j. We also conjecture that for any positive integer m there are consecutive primes p_k,...,p_n (k

Motivation & Objective

• To construct number-theoretic functions whose outputs are exclusively prime numbers using modular distinctness conditions.
• To characterize the least modulus $m > 1$ such that sequences like $\binom{2k}{k}$, $k!$, $k(k-1)$, or $2k(k-1)$ are pairwise distinct modulo $m$, and show that this modulus is always prime.
• To extend prior work on $D_f(n)$ by proving that for specific polynomials $f$, the minimal modulus ensuring distinct values modulo $m$ yields a prime or prime power.
• To propose and support conjectures linking linear recurrence sequences and primitive roots to prime outputs via minimal distinctness moduli.

Proposed method

• Define $S(n)$ as the smallest $m > 1$ such that $2k(k-1) \mod m$ are pairwise distinct for $k = 1, \dots, n$, and prove $S(n)$ equals the least prime greater than $2n-2$.
• Define $T(n)$ as the smallest $m > 1$ such that $k(k-1) \mod m$ are pairwise distinct for $k = 1, \dots, n$, and prove $T(n) = \min\{m \geq 2n-1 : m \text{ is prime or a power of } 2\}$.
• Use properties of quadratic residues, Legendre symbols, and recurrence sequences (e.g., Lucas sequences) to analyze distinctness modulo $m$ and link them to primality.
• Prove that for $f(k) = k(dk - 1)$, the minimal $m$ ensuring $n$ distinct values modulo $m$ is the least power of $d$ not smaller than $n$, for $d \in \{2,3\}$.
• Analyze sequences involving Euler numbers $E_{2k}$ and show that the minimal modulus for distinctness of $E_{2k} \mod m$ is a power of two or a square of a prime, with few exceptions.
• Use Stern's congruence and properties of Lucas sequences to derive conditions under which $v_k(A,1) \mod m$ are distinct, leading to conjectures on primality of minimal moduli.

Experimental results

Research questions

• RQ1 Can a function based on the minimal modulus ensuring distinct values of a polynomial sequence modulo $m$ produce only primes?
• RQ2 Is the smallest $m > 1$ such that $2k(k-1) \mod m$ are pairwise distinct always prime, and if so, which prime?
• RQ3 For sequences like $k(k-1) \mod m$, does the minimal $m$ that ensures $n$ distinct values equal the smallest prime or power of two at least $2n-1$?
• RQ4 Do minimal moduli for distinctness of $k(dk-1) \mod m$ yield powers of $d$, and under what conditions?
• RQ5 Can the minimal modulus ensuring distinct values of $E_{2k} \mod m$ be characterized, and when is it prime?

Key findings

• The function $S(n)$, defined as the smallest $m > 1$ such that $2k(k-1) \mod m$ are pairwise distinct for $k = 1, \dots, n$, equals the least prime greater than $2n - 2$.
• The function $T(n)$, defined as the smallest $m > 1$ such that $k(k-1) \mod m$ are pairwise distinct for $k = 1, \dots, n$, equals the smallest integer $m \geq 2n - 1$ that is either prime or a power of 2.
• For $d \in \{2,3\}$, the minimal $m$ such that $k(dk - 1) \mod m$ are pairwise distinct is the least power of $d$ not smaller than $n$, i.e., $d^{\lceil \log_d n \rceil}$.
• For $n \geq 4$, the minimal $m$ such that $18k(3k-1) \mod m$ are pairwise distinct is the least prime $p > 3n$ with $p \equiv 1 \pmod{3}$.
• The minimal $m$ such that $4k(4k-1) \mod m$ are pairwise distinct for $k = 1, \dots, n$ (with $n \geq 5$) is the least prime $p > (8n - 4)/3$ with $p \equiv 1 \pmod{4}$.
• For $n \geq 6$, the minimal $m$ such that $4k(4k+1) \mod m$ are pairwise distinct is the least prime $p > (8n - 2)/3$ with $p \equiv -1 \pmod{4}$.